# Monday, June 9

09:00-09:50 |
Laurent Younes (Johns Hopkins Univ.): Metamorphosis: A General Framework for Metric Registration |

09:50-10:40 |
Joan Glaunès (Université René Descartes): Using Vectorial Splines for Large Deformations and Surface Matching |

10:40-11:10 |
Coffee Break |

11:10-11:40 |
Stanley Durrleman (INRIA-project-team Asclepios (Sophia Antipolis Cedex)): Sparse Approximation of Currents for Statistics on Curves and Surfaces |

11:40-12:10 |
Francois Xavier Vialard (ENS de Cachan): From the Hamiltonian Formulation of Image Matching to a General Shape Diffusion Model |

12:10-15:00 |
Lunch Break - Discussion |

15:00-15:50 |
Washington Mio (Florida State University): Shape Spaces of Manifolds and Simplicial Complexes |

15:50-16:20 |
Thomas Schoenemann (Universität Bonn): Shape Optimization in Computer Vision via Minimal Cycles in Graphs |

16:20-16:50 |
Coffee Break |

16:50-17:20 |
Frank R. Schmidt (Universität Bonn): Efficient Methods for Measuring the Dissimilarity of Planar Shapes |

17:20-19:00 |
BMW (Pretzel Mathematics Wine) |

# Tuesday, June 10

09:00-09:50 |
Denis Zorin (New York University):Numerical Simulation of Fluid Membranes |

09:50-10:40 |
Ron Kimmel, (Israel Institut of Technology (Technion)):Numerical Geometry of Nonrigid Shapes |

10:40-11:10 |
Coffee Break |

11:10-12:00 |
Guillermo Sapiro (University of Minnesota): Bending Invariant Representation and Manipulation of Shapes |

12:00-15:00 |
Lunch Break - Discussion |

15:00-15:50 |
Jayant Shah (Northeastern University): Sobolev Metrics on the Space of Planar Curves Modulo Similitudes |

15:50-16:20 |
Paolo Favaro (Heriot Watts University): A Modified Gradient-Flow with Application to Shape from Defocus |

16:20-16:50 |
Coffee Break |

16:50-17:20 |
Patrick Degener (Universität Bonn): Automatic Design of Panoramic Maps |

17:20-17:50 |
Stephan Huckemann (Universität Göttingen): Intrinsic Shape Analysis: Geodesic PCA for Riemannian Manifolds Modulo Isomorphic Lie Group Actions |

# Wednesday, June 11

09:00-09:50 |
Peter Schröder (California Institute of Technology): Discrete Conformal Equivalence of Triangle Meshes |

09:50-10:40 |
Peter Giblin (University of Liverpool): Views of Illuminated Surfaces |

10:40-11:10 |
Coffee Break |

11:10-11:40 |
Matt Feiszli, (Brown University): Representation and Compression of 2-D Shapes Using Conformal Mapping |

11:40-12:10 |
Folkmar Bornemann (Technische Universität München):Modified Distance Maps for Image Inpainting |

12:10-15:00 |
Lunch Break - Discussion |

15:00-15:50 |
Antonin Chambolle (Ecole Polytechnique): On the Limit of Some Nonlocal Perimeters |

15:50-16:20 |
Kalin Kolev (Universität Bonn): Convex Shape Optimization in Multiview 3D Reconstruction |

16:20-16:50 |
Coffee Break |

16:50-17:20 |
Benedikt Wirth (Universität Bonn): Mumford-Shah-Based Elastic Shape Averaging |

17:20-17:50 |
Leah Bar (University of Minnesota): Generalized Newton Method for Energy Formulations in Image Processing |

# Thursday, June 12

09:00-09:50 |
Mads Nielsen (IT University of Copenhagen): Brownian Warps for Non-Rigid Image Registration A Brownian |

09:50-10:40 |
Nir Sochen (University of Tel Aviv): New Transform for Parameterized 2D Shapes |

10:40-11:10 |
Coffee Break |

11:10-12:00 |
Tom Fletcher (University of Utah): Robust Statistics on Riemannian Manifolds via the Geometric Median |

12:00-15:00 |
Lunch Break - Discussion |

15:00-15:50 |
Otmar Scherzer (University of Innsbruck): Regularized Reconstruction of Shapes with Statistical A Priori Knowledge |

15:50-16:20 |
Michael Wand (Max-Planck-Institut für Informatik (Saarbrücken)):Animation Reconstruction |

16:20-16:50 |
Coffee Break |

16:50-17:20 |
Florian Steinke (Max-Planck Insitut für Biological Cybernetics (Tübingen)): Thin-plate Splines between Riemannian Manifolds |

17:20-17:50 |
Arjan Kuijper (RICAM Austrian Academy of Science (Linz)): 2D Shape Matching using Symmetry Sets |

19:00 |
Conference Dinner |

# Friday, June 13

09:00-09:50 |
Giovanni Bellettini (Universita’ di Roma "Tor Vergata"): On the Reconstruction of a Solid Shape from its Apparent Contour |

09:50-10:40 |
Sarang Joshi (University of North Carolina at Chapel Hill): Statistics of Shape: Simple Statistics on Interesting Spaces |

10:40-11:11 |
Coffee Break |

11:10-12:00 |
Peter Michor (Universitaet Wien): Shape Spaces and their Curvatures |

12:00-12:30 |
Mario Micheli (Brown University): On the Curvature of Landmarks Manifolds |

12:30-15:00 |
Lunch Break - Discussion |

## Laurent Younes (Johns Hopkins Univ.): Metamorphosis: A General Framework for Metric Registration

In the pattern matching approach to imaging science, we introduce the process of metamorphosis, which can be interpreted as "template matching with dynamical templates". Metamorphosis allows for a simultaneous registration and metric comparison between various types of mathematical representation of shapes. We will develop a general theory for this framework, involving the action of groups on Riemannian manifolds; this will provide the relevant equations and variational problems. We will show applications of this theory to landmark, image and density matching.

This is joint work with Darryl Holm and Alain Trouve.

## Joan Glaunès (Université René Descartes): Using Vectorial Splines for Large Deformations and Surface Matching Using Vectorial Splines for Large Deformations and Surface Matching

Large deformations methods in image analysis have received large interest in the past recent years. The usual theoretical construction starts from a Hilbert norm on vector fields that controls the regularity of the infinitesimal transformations, and which determines the class of deformations. In many cases, more precisely in the case of sparse data analysis, such as points, curves or surfaces, the central object on which algorithms rely is the reproducing kernel of the corresponding Hilbert space. The choice of kernel: gaussian, scalar or vectorial, or semi-kernel, and its parameters, is essential, and we will see how it affects the type of deformations one can get. Reproducing Kernel Hilbert Spaces are also convenient tools when modeling curves and surfaces via vector-valued measures (currents). In this setting also, one can get interesting new behaviours by using exotic kernel types.

## Stanley Durrleman (INRIA-project-team Asclepios (Sophia Antipolis Cedex)): Sparse Approximation of Currents for Statistics on Curves and Surfaces

Computing, processing, visualizing statistics on shapes like curves or surfaces is a real challenge with many applications ranging from medical image analysis to computational geometry. Modelling such geometrical primitives with currents avoids feature-based approach as well as point-correspondence method. This framework has been proved to be powerful to register brain surfaces or to measure geometrical invariants. However, if the state-of-the-art methods perform efficiently pairwise registrations, new numerical schemes are required to process groupwise statistics due to an increasing complexity when the size of the database is growing. Statistics such as mean and principal modes of a set of shapes often have a heavy and highly redundant representation. We propose therefore to find an adapted basis on which mean and principal modes have a sparse decomposition. Besides the computational improvement, this sparse representation offers a way to visualize and interpret statistics on currents. Experiments show the relevance of the approach on 34 sets of 70 sulcal lines and on 50 sets of 10 meshes of deep brain structures.

## Francois Xavier Vialard (ENS de Cachan): From the Hamiltonian Formulation of Image Matching to a General Shape Diffusion Model

We will extend the framework of image matching through a diffeomorphism approach to discontinuous images which are considered as function of bounded variations (more precisely SBV functions). We will give in this framework an Hamiltonian formulation of the geodesic equations (which generalizes the landmark case) for which we can prove uniqueness and existence results. Last, we will present a shape diffusion model which is currently studied.

## Washington Mio (Florida State University): Shape Spaces of Manifolds and Simplicial Complexes

Washington Mio (Florida State University): Shape Spaces of Manifolds and Simplicial Complexes We describe the construction of shape spaces of manifolds and simplicial complexes equipped with a family of geodesic metrics indexed by elasticity parameters that characterize the tension and rigidity of the shapes. The general model is anisotropic and inhomogeneous and is developed within the framework of Riemannian geometry. Although geodesic distance is a global quantifier of shape dissimilarity, the geodesic deformation fields allow us to define energy density functions that lend a local-global character to the model by letting us quantify the local contributions to shape similarity and divergence. We present applications to brain mapping, such as the construction of anatomical atlases of the hippocampus and the cortex. Time permitting, we will also discuss a simpler parametric model based on Sobolev metrics, including several illustrations.

## Thomas Schoenemann (Universität Bonn): Shape Optimization in Computer Vision via Minimal Cycles in Graphs

In this talk, I will show that a number of difficult (non-convex) shape optimization problems arising in Computer Vision can be solved in a globally optimal manner by formulating them as minimal cycle problems in appropriate graphs. In particular, I will show how to impose curvature regularity in image segmentation and how to optimally match deformable shapes to images.

## Frank R. Schmidt (Universität Bonn): Efficient Methods for Measuring the Dissimilarity of Planar Shapes

In the first part of my talk, I will introduce a variational method to compute geodesics on a manifold of shapes. I will show that in contrast to the commonly used shooting method it provides distances which are symmetric and numerically more stable to compute.

In the second part of my talk I will discuss methods to directly compute a shape dissimilarity without computing trajectories on a manifold. In these approaches, the problem of finding the globally optimal diffeomorphism between two shapes is cast as a problem of finding a minimal cycle on a torus. I will present an algorithm which we believe to be the currently most efficient method to solve this problem.

## Denis Zorin (New York University):Numerical Simulation of Fluid Membranes

Fluid membranes are area-preserving interfaces that resist bending. They models of cell membranes, intracellular organelles, and viral particles. I will discuss algorithms for efficient and accurate simulation of suspensions of deformable vesicles in Stoksean fluids based on boundary integral equations. This type of techniques entirely avoids discretization of the fluid domain, and allows to compute membrane deformations with high accuracy using a small number of degrees of freedom for each particle. Our approach requires a combination of high-order discretization for the boundaries supporting accurate singular quadratures efficient and scalable techniques for solving dense linear systems resulting from discretization of integral equations and efficient stable time-stepping. I will primarily focus on the planar case, with a brief overview of the state of the art in 3D.

Based on joint work with Lexing Ying, Denis Gueyffier, Shravan K. Veerapaneni and George Biros.

## Ron Kimmel, (Israel Institut of Technology (Technion)):Numerical Geometry of Nonrigid Shapes

In this lecture I will review some recent results emerging in a new field we named “Numerical Geometry of Nonrigid Shapes.” I will start with a brief motivation from biometry, in which shapes we deal with can be approximated as almost isometric structures. This is the case for example for 3D face recognition. We explore various measures for the goal of matching, comparing, interpolating, and finding the correspondence between various shapes for which almost-isometry can be assumed. Next, we extend those concepts into intrinsic symmetries, elaborate on partial matching, and comment on the relation of the measures we use to known results from the theory of metric geometry and geometry of curved manifolds. Finally, I will comment on extensions of the the underlying principles to manifold learning and nonlinear dimensionality reduction.

## Guillermo Sapiro (University of Minnesota): Bending Invariant Representation and Manipulation of Shapes

Based on the theory of Gromov-Hausdorff distances and Lipschitz minimizing flow, we will present some theoretical and computational results on bending invariant shape recognition and warping. We will also describe a distributions based system for state-of-the-art recognition of point cloud 3D objects.

## Jayant Shah (Northeastern University): Sobolev Metrics on the Space of Planar Curves Modulo Similitudes

An H^{1}-metric on the space of planar curves modulo similitudes has been analyzed in detail in terms of the geodesics and the sectional curvature by Younes et al. In this talk, I will discuss Sobolev metrics of higher order on this space. The H^{2} metric may be of particular interest for practical applications. I will describe an approach for analyzing general H^{n} metrics and their behavior as n tends to infinity.

## Paolo Favaro (Heriot Watts University): A Modified Gradient-Flow with Application to Shape from Defocus

In many estimation problems one has to simultaneously infer dishomogeneous quantities from images. For instance, in shape from defocus one needs to infer shape (a geometrical quantity) and texture (a photometrical quantity). In this case, if one poses the inference problem as an energy minimization, it is possible to match the domain of only one unknown at a time. While this has virtually no effect on noiseless data, in the presence of noise this results in an undesired regularization on the unmatched unknown. To address this problem we propose a novel regularized iterative method based on a modified gradient flow. This flow is possibly not the gradient of any energy functional, and provides a different tradeoff between the minima of the data term and the minima of the regularization term. We show how this can be applied to shape from defocus and how this improves the reconstruction of shape and texture in both synthetic and real images.

## Patrick Degener (Universität Bonn): Automatic Design of Panoramic Maps

Panoramic maps combine the advantages of both ordinary geographic maps and terrestrial images. While inheriting the familiar perspective of terrestrial images, they provide a good overview and avoid occlusion of important geographical features. The designer achieves this by skillful choice and integration of several views in a single image. As important features on the surface must be carefully rearranged to guarantee their visibility, the manual design of panoramic maps requires many hours of tedious and painstaking work. In this paper we take a variational approach to the design of panoramic maps. Starting from conventional elevation data and aerial images our method fully automatically computes panoramic maps from arbitrary viewpoints. It rearranges geographic structures to maximize the visibility of a specified set of features while minimizing the deformation of the landscape’s shape.

## Stephan Huckemann (Universität Göttingen): Intrinsic Shape Analysis: Geodesic PCA for Riemannian Manifolds Modulo Isomorphic Lie Group Actions

Many shape spaces are orbit spaces of a Riemannian manifold modulo an isormphic Lie group action. Classically, these non-Euclidean spaces are approximated by linear Euclidean spaces, thus making standard multivariate techniques applicable. In particular for 3D applications however, this "extrinsic" approximation often may be non-valid. Hence, basic statistical descriptors such as means, variance and its decomposition into principal components need to be defined "intrinsically". For such non-Euclidean quotient spaces, frameworks for intrinsic PCA are discussed and a specific approach is proposed. This method can be applied to wide spread data as well as for data in high curvature regions.

In collaboration with Thomas Hotz and Axel Munk

## Peter Schröder (California Institute of Technology): Discrete Conformal Equivalence of Triangle Meshes

We present a new algorithm for conformal mesh parameterization. It is based on a precise notion of discrete conformal equivalence for triangle meshes which mimics the notion of conformal equivalence for smooth surfaces. The problem of finding a flat mesh that is discretely conformally equivalent to a given mesh can be solved efficiently by minimizing a convex energy function, whose Hessian turns out to be the well known cot-Laplace operator. This method can also be used to map a surface mesh to a parameter domain which is flat except for isolated cone singularities, and we show how these can be placed automatically in order to reduce the distortion of the parameterization. We present the salient features of the theory and elaborate the algorithms with a number of examples.

## Peter Giblin (University of Liverpool): Views of Illuminated Surfaces

There are clues to the shape of an object contained in the movement of surface markings, shadows, "shade curves" (terminators) and apparent contours, as the observer moves past the object. There have been a number of attempts to codify all these interactions, using singularity theory, but some of these have been flawed in that they use "results" which, in the context, are not true. I shall talk about some recent work with James Damon (UNC Chapel Hill) and my former postdoc Gareth Haslinger which is a systematic attempt to classify all the possible interactions. The talk will not attempt to be comprehensive but will concentrate on some particular cases of interest. Two papers, one in draft form, are available on my webpage www.liv.ac.uk/~pjgiblin and others are in preparation.

## Matt Feiszli, (Brown University): Representation and Compression of 2-D Shapes Using Conformal Mapping

The theory of Teichmuller spaces provides an isomorphism between certain normalized diffeomorphisms of the circle and smooth simple closed plane curves. Thus, for any smooth shape there is an associated "fingerprint": a self-map of the circle. Through careful study of the boundary derivatives of conformal maps, we demonstrate that this fingerprint encodes a sort of continuous version of the medial axis of the shape. In addition, we present explicit estimates showing that conformal maps encode the local Euclidean curvature of the boundary. The machinery involved in making our geometric estimates comes both from classical function theory and hyperbolic geometry. We next describe a family of algorithms for compressing realvalued functions defined on an interval. Via the isomorphism between self-maps of the circle, viewed as maps of the interval [0,2*pi] into itself, and simple closed plane curves provided by Teichmuller theory, this gives rise to a family of compressors for 2-dimensional shapes. We’ll describe the way in which these compressors work and how the geometry of a shape determines the number and allocation of bits used to represent it.

## Folkmar Bornemann (Technische Universität München):Modified Distance Maps for Image Inpainting

The geometric interplay between the distance map and the edge-enhancing coherence flow field is constitutive for the image quality of the fast inpainting method of Bornemann and März. We sketch the implications of a recent existence proof for the nonlinear limit model and discuss some options to modify the distance map for improved performance of the method.

## Antonin Chambolle (Ecole Polytechnique): On the Limit of Some Nonlocal Perimeters

In this talk, I will consider discrete approximations of the perimeters, that are easily and efficiently minimized by combinatorial optimization techniques. The point is to investigate whether it is interesting or not to consider interactions involving more than two points, to improve the quality of the approximation. We show a general convergence result for such energies, from which it seems to appear that more-than-two-points interactions never do better than two-points interactions, as long as one requires the submodularity of the approximation.

This is a joint work with A. Giacomini and L. Lussardi.

## Kalin Kolev (Universität Bonn): Convex Shape Optimization in Multiview 3D Reconstruction

In this talk, I will show how the classical computer vision problem of reconstructing 3D shape from a collection of 2D images can be cast as one of minimizing a continuous convex functional representing a stereo-weighted minimal surface problem. Since the subdivision of 3D space into interior and exterior involves an optimization over the non-convex set of binary labelings, we make use of recently proposed convex relaxation methods. For the functionals we consider thresholding of the relaxed solution provides a global optimum of the binary problem. In addition, we prove that exact silhouette-consistency can be imposed as a convex constraint.

## Benedikt Wirth (Universität Bonn): Mumford-Shah-Based Elastic Shape Averaging

A method is presented to compute an average representation of a given number of shapes. The method is based on image registration via edge matching and uses a hyperelastic regularization. The corresponding model is stated in a variational form and implemented as a fixed point iteration of gradient descent steps, using finite element methods in a multilevel framework. Results are presented that show the model’s applicability to finding e. g. the average shape of a human organ or even to produce simple anatomical template maps.

## Leah Bar (University of Minnesota): Generalized Newton Method for Energy Formulations in Image Processing

Many problems in image processing are addressed via the minimization of a cost functional. The most prominent optimization technique is the gradient descent, often used due to its simplicity and applicability where other techniques, e.g., those coming from discrete optimization, can not be used. Yet, gradient descent suffers from a slow convergence, and often to just local minima which highly depends on the condition number of the functional Hessian. Newton-type methods, on the other hand, are known to have a rapid, quadratic, convergence. In its classical form, the Newton method relies on the L^{2}-type norm to define the descent direction. In this paper, we generalize and reformulate this very important optimization method by introducing a novel Newton method based on more general norms. This generalization opens up new possibilities in the extraction of the Newton step, including benefits such as mathematical stability and the incorporation of smoothness constraints. We first present the derivation of the modified Newton step in the calculus of variation framework needed for image processing. Then, we demonstrate the method with two common objective functionals: variational image deblurring and geodesic active contours for image segmentation. We show that in addition to the fast convergence, norms adapted to the problem at hand yield different and superior results.

## Mads Nielsen (IT University of Copenhagen): Brownian Warps for Non-Rigid Image Registration A Brownian

A Brownian motion model in the group of diffeomorphisms has been introduced as inducing a least committed prior on warps. This prior is source-destination symmetric, fulfills a natural semi-group property for warps, and with probability 1 creates invertible warps. Using this as a least committed prior, we formulate a Partial Differential Equation for obtaining the maximally likely warp given matching constraints derived from the images. We demonstrate the technique on 2D images, and show that the obtained warps are also in practice source-destination symmetric and in an example on X-ray spine registration provides extrapolations from landmark point superior to those of spline solutions.

## Nir Sochen (University of Tel Aviv): New Transform for Parameterized 2D Shapes

We show how representation theory and the Heisenberg group can be combined to represent parameterized shapes in a new way. The new transform is a generalization of the discrete Fourier transform.

## Tom Fletcher (University of Utah): Robust Statistics on Riemannian Manifolds via the Geometric Median

The geometric median is a classic robust estimator of centrality for data in Euclidean spaces. In this talk I will present a formulation of the geometric median of data on a Riemannian manifold as the minimizer of the sum of geodesic distances to the data points. We prove existence and uniqueness of the geometric median on manifolds with non-positive sectional curvature and give sufficient conditions for uniqueness on positively curved manifolds. Generalizing the Weiszfeld procedure for finding the geometric median of Euclidean data, we present an algorithm for computing the geometric median on an arbitrary manifold. We show that this algorithm converges to the unique solution when it exists. This method produces a robust central point for data lying on a manifold, and should have use in a variety of vision applications involving manifolds. I’ll show examples of the geometric median computation and demonstrate its robustness for three types of manifold data: the 3D rotation group, tensor manifolds, and shape spaces.

## Otmar Scherzer (University of Innsbruck): Regularized Reconstruction of Shapes with Statistical A Priori Knowledge

The reconstruction of geometry or, in particular, the shape of objects is a common issue in image analysis. Starting from a variational formulation of such a problem on a shape manifold we introduce a regularization technique incorporating statistical shape knowledge. The key idea is to consider a Riemannian metric on the shape manifold which reflects the statistics of a given training set. We investigate the properties of the regularization functional and illustrate our technique by applying it to region-based and edge-based segmentation of image data. In contrast to previous works our framework can be considered on arbitrary (finite-dimensional) shape manifolds and allows the use of Riemannian metrics for regularization of a wide class of variational problems in image processing.

This is joint work with Matthias Fuchs, Innsbruck.

## Michael Wand (Max-Planck-Institut für Informatik (Saarbrücken)):Animation Reconstruction

The talk discusses the reconstruction of animations from real-time 3D scanner data. The goal of this work is to reconstruct an object and its deformation over time from a spatio-temporal sample of surface points. We have assume that these measurements are corrupted by noise and outliers and in particular, we also have to deal with the problem that the scanning device will in each frame only be able to acquire an incomplete, partial view of the object. The talk will present a new technique that solves this reconstruction problem. The approach uses statistical priors on shape and motion to estimate a completed object and its trajectory over time. The talk will address the statistical model, numerical and discrete optimization techniques for the estimation, and show some results for synthetic and real-world data sets.

## Florian Steinke (Max-Planck Insitut für Biological Cybernetics (Tübingen)): Thin-plate Splines between Riemannian Manifolds

With the help of differential geometry we describe a framework to define a thin-plate spline like energy for maps between arbitrary Riemannian manifolds. The so-called Eells energy only depends on the intrinsic geometry of the input and output manifold, but not on their respective representation. The energy can then be used for regression between manifolds, we present results for cases where the outputs are rotations, sets of angles, or points on 3D surfaces. In the future we plan to also target regression where the output is an element of "shape space", understood as a Riemannian manifold. One could also further explore the meaning of the Eells energy when applied to diffeomorphisms between shapes, especially with regard to its potential use as a distance measure between shapes that does not depend on the embedding or the parametrisation of the shapes.

Initial publication: Steinke, F., M. Hein, J. Peters and B. Schölkopf: Manifold-valued Thin-Plate Splines with Applications in Computer Graphics. Computer Graphics Forum 27(2, EUROGRAPHICS 2008) (04 2008)

## Arjan Kuijper (RICAM Austrian Academy of Science (Linz)): 2D Shape Matching using Symmetry Sets

In this talk I will give an overview of my work on 2D Shape Matching using Symmetry Sets.

The Symmetry Set of a simple 2D shape is defined as the closure of the mid points of circles tangent to a shape at at least two places. A well-known subset, called the Medical Axis, is obtained by restricting to the set of maximal circles.

I will describe some properties of the Symmetry Set and its use as shape descriptor. An alternative descriptor is given by the set of point pairs (and triples) on the shape defining the circles.

When a shape is gradually changed, the Symmetry Set will go through transitions (or singularities), which are well-understood. These transitions can be transferred to the shape descriptors. A sequence of transitions can be regarded as changing one shape into another. We designed algorithms for shape matching that are based on these transitions. Examples will be shown during the talk.

This work was carried out in collaboration with Peter Giblin (Liverpool), Ole Fogh Olsen, and Philip Bille (both Copenhagen). Several papers describing various aspects of the work can be found on www.ricam.oeaw.ac.at/people/page/kuijper/publications.html

## Giovanni Bellettini (Universita’ di Roma "Tor Vergata"): On the Reconstruction of a Solid Shape from its Apparent Contour

We discuss a two-dimensional variational model for the reconstruction of a smooth generic solid shape, which may handle self-occlusions. The model is based on the Huffman labelling of the apparent contour. Such a labelling is the starting point to answer to the problem of recovering a three-dimensional layered shape from its apparent contour. Connections of the model with the completion of hidden contours are also discussed.

## Sarang Joshi (University of North Carolina at Chapel Hill): Statistics of Shape: Simple Statistics on Interesting Spaces

A primary goal of Computational Anatomy is the statistical analysis of anatomical variability. A natural question that arises is how dose one define the image of an “Average Anatomy”. Such an “average” must represent the intrinsic geometric anatomical variability present. Large Deformation Diffeomorphic transformations have been shown to accommodate the geometric variability but performing statistics of Diffeomorphic transformations remains a challenge.

In this lecture I will further extend this notion of averaging for studying change of anatomy on average from a cross sectional study of growth. Regression analysis is a powerful tool for the study of changes in a dependent variable as a function of an independent repressor variable, and in particular it is applicable to the study of anatomical growth and shape change. When the underlying process can be modeled by parameters in a Euclidean space, classical regression techniques are applicable and have been studied extensively.

However, recent work suggests that attempts to describe anatomical shapes using flat Euclidean spaces undermines our ability to represent natural biological variability. In this lecture I will develop a method for regression analysis of general, manifold-valued data. Specifically, we extend Nadaraya-Watson kernel regression by recasting the regression problem in terms of Fréchet expectation. Although this method is quite general, our driving problem is the study anatomical shape change as a function of age from random design image data.

I will demonstrate our method by analyzing shape change in the brain from a random design dataset of MR images of 97 healthy adults ranging in age from 20 to 79 years. To study the small scale changes in anatomy, we use the infinite dimensional manifold of diffeomorphic transformations, with an associated metric. We regress a representative anatomical shape, as a function of age, from this population.

## Peter Michor (Universitaet Wien): Shape Spaces and their Curvatures

The talk will concentrate on shape spaces as homogeneous spaces under the diffeomorphism group, like landmark spaces, spaces of currents, etc. The talk will try to give the sectional curvatures for them: directly, and via O’Neill’s formulas.

## Mario Micheli (Brown University): On the Curvature of Landmarks Manifolds

It is well known that Shape Spaces can be endowed with the structure of Riemannian manifold; this allows one to compute, for example, Euler-Lagrange equations and geodesic distance for such spaces. However until very recently little was known about the actual geometry of Shape Manifolds, e.g. about their curvature. In this talk we summarize some recent results on the curvature of one of the simplest Shape Manifolds, namely the one of landmark points. Besides showing the analytic expressions for such curvature in several cases, we will also describe the effect of curvature on the qualitative dynamics of landmarks and discuss possible implications on the statistical analysis of shapes.